Nonsymmorphic symmetry-required band crossings in topological semimetals

Y. X. Zhao1, ∗ and Andreas P. Schnyder1, †

1Max-Planck-Institute for Solid State Research, D-70569 Stuttgart, Germany(Dated: October 13, 2016)

We show that for two-band systems nonsymmorphic symmetries may enforce the existence of bandcrossings in the bulk, which realize Fermi surfaces of reduced dimensionality. We find that theseunavoidable crossings originate from the momentum dependence of the nonsymmorphic symmetry,which puts strong restrictions on the global structure of the band configurations. Three differenttypes of nonsymmorphic symmetries are considered: (i) a unitary nonsymmorphic symmetry, (ii) anonsymmorphic magnetic symmetry, and (iii) a nonsymmorphic symmetry combined with inversion.For nonsymmorphic symmetries of the latter two types, the band crossings are located at high-symmetry points of the Brillouin zone, with their exact positions being determined by the algebraof the symmetry operators. To characterize these band degeneracies we introduce a global topologicalcharge and show that it is of Z2 type, which is in contrast to the local topological charge of Fermipoints in, say, Weyl semimetals. To illustrate these concepts, we discuss the π-flux state as wellas the SSH model at its critical point and show that these two models fit nicely into our generalframework of nonsymmorphic two-band systems.

PACS numbers: 03.65.Vf, 71.20.-b, 73.20.-r,71.90.+q

I. INTRODUCTION

Since the experimental discovery of topological insula-tors1,2, symmetry protected topological phases have be-come a major research subject3–8. Recent studies havebeen concerned with topological phases that are pro-tected by spatial symmetries, such as topological crys-talline insulators9–12 and topological semimetals stabi-lized by reflection, inversion, or other crystal symme-tries13–15. Until recently, the study of these topologi-cal crystalline materials has focused on the role of pointgroup symmetries. However, besides point group symme-tries the space group of a crystal can also contain non-symmorphic symmetries, which are combinations of pointgroup operations with nonprimitive lattice translations.It has been shown that the presence of nonsymmorphicsymmetries leads to new topological phases, which canbe insulating16–21, or semimetallic with Dirac points pro-tected by nonsymmorphic symmetries22,23. In the lattercase, the Dirac points possess local topological charges,which guarantees their local stability.

However, as we show in this paper, nonsymmorphicsymmetries restrict the form of the band structure notonly locally but also globally, which may lead to unavoid-able band crossings in the bulk24–28. Indeed, the nonsym-morphic symmetries can put so strong constraints on theglobal properties of the band structure that the systemis required by symmetry to be in a topological semimetalphase, with Fermi surfaces of reduced dimensionality.These symmetry-enforced semimetals possess low-energyexcitations with unconventional dispersions and may ex-hibit novel topological response phenomena and unusualmagneto-transport properties. In the following we con-sider three different types of nonsymmorphic symmetries:(i) Unitary nonsymmorphic symmetries, (ii) nonsymmor-phic symmetries combined with inversion, and (iii) non-symmorphic magnetic symmetries. We first rigorously

prove that for any one-dimensional (1D) two-band systemunitary nonsymmorphic symmetries enforce the existenceof band crossings, due to global topological constraintson the band structure. In the presence of an additionalinversion symmetry, the symmetry enforced band degen-eracies are located either at the origin or at the boundaryof the Brillouin zone (BZ), depending on the algebra ofthe symmetry operators. The same holds true for non-symmorphic magnetic symmetries, which are composedof a unitary nonsymmorphic symmetry followed by ananti-unitary time-reversal symmetry. We present gener-alizations of these results to higher dimensions, for whichnonsymmorphic symmetries may enforce the existence ofzero- or higher-dimensional band crossings. In all of theabove cases we find that the nonsymmorphic symmetriesrestrict the momentum space structure in the BZ both lo-cally and globally. To characterize the global topologicalfeatures we introduce a novel global topological charge,which as we show, is always of Z2 type. Hence, the globaltopological features exhibit a Z2 classification, which isin contrast to the local topological characteristics, whichpossess a Z classification. Finally, we illustrate thesefindings by considering two prototypical examples: (i)the π-flux square lattice model and (ii) the SSH modelat its critical point. Within our unified framework, weshow that the former model can be viewed as the higher-dimensional generalization of the latter.

II. UNITARY NONSYMMORPHIC SYMMETRY

We start by considering a general 1D two-band Hamil-tonian H(k) with the two-fold unitary nonsymmorphicsymmetry

G(k) =

(0 e−ik

1 0

), (1)

arX

iv:1

606.

0369

8v2

[co

nd-m

at.m

es-h

all]

11

Oct

201

6

2

FIG. 1: Illustration of nonsymmorphic symmetries in one-dimensional lattices. (a) The nonsymmorphic symmetry iscomposed of a π rotation followed by a half translation a/2,where a is the lattice constant. (b) The nonsymmorphic mag-netic symmetry is composed of the two operations of (a) fol-lowed by the exchange of black and white balls, which repre-sents time-reversal symmetry.

which acts on H(k) as

G(k)H(k)G−1(k) = H(k). (2)

Since G2(k) = e−ikσ0, the eigenvalues of G(k) are±e−ik/2. Therefore, the nonsymmorphic symmetry G(k)can be viewed as an operation on internal degrees of free-dom (e.g., pseudospin) followed by a half translation,as illustrated in Fig. 1(a). Observing that G(k) anti-commutes with σ3, the Hamiltonian can be written as

H(k) =

(0 q(k)

q∗(k) 0

). (3)

Without loss of generality, we have dropped the termproportional to the identity, which only shifts the energyof eigenstates. Inserting Eqs. (3) and (1) into Eq. (1), wefind that due to the non-symmorphic symmetry G(k),q(k) must satisfy

q(k)eik = q∗(k). (4)

We claim that any periodic function q(k) satisfyingEq. (4) has zeros, and thus any two-band model withthe nonsymmorphic symmetry (1) is required to be gap-less. To see this, we introduce f(z) = q(k) with z = eik,from which it follows that zf(z) = f∗(z). If q(k) or f(z)is nonzero on the unit circle S1, then

z = f∗(z)/f(z), (5)

which however is impossible. This is because for z ∈ S1,the two sides of Eq. (5) both define functions from S1

to S1, but the left-hand side has winding number 1,while the winding of the right-hand side is even, sincef∗(z)/f(z) = e2iArc[f(z)]. Thus, q(k) must vanish at somemomentum by contradiction. For the topological argu-ment to work for multi-band theories, we may replaceq(k) in Eq.(3) by the determinant of the off-diagonal en-try, which is discussed in Sec.VII.

III. NONSYMMORPHIC SYMMETRYCOMBINED WITH INVERSION SYMMETRY

We note that while a unitary nonsymmorphic symme-try guarantees the existence of a band crossing point,it does not fix the position of this degeneracy pointin momentum space. However, in the presence of anadditional inversion symmetry, the band crossings arepinned to either the origin or the boundary of the BZ. Todemonstrate this, let us consider the inversion symmetryP = σ2ı, where ı inverses the momentum. We find that

[H, P ] = 0, PG(k)P−1 = −GT (−k), P 2 = −1. (6)

Since q(k) is a periodic function, we expand it as q(k) =∑n qne

ink. It follows from Eq. (4) that q−(n+1) = q∗n,which, as a recursion relation, allows us to express q(k)as

q(k) =∞∑n=0

(qneink + q∗ne

−i(n+1)k). (7)

From Eq. (6) it follows that σ2H(−k)σ2 = H(k), whichimplies q(k) = −q∗(−k) or equivalently qn = −q∗n. Sinceqn are all purely imaginary, we find that

q(k) =

∞∑n=0

λni

(eink − e−i(n+1)k), (8)

with λn being real numbers. We observe that indepen-dent of λn there always exists a band crossing point atk = 0. For example, by keeping only the zeroth term inEq. (8), one finds as a simple concrete model,

H0(k) = λ sin kσx + λ(1− cos k)σy. (9)

We note that the nonsymmorphic symmetry G(k) re-lates seemingly independent terms to each other in theHamiltonian. This is exemplified by Eq. (9), whereall three terms (which are usually independent) havethe same coefficients. Obviously, higher order terms inEq. (8), which constitute symmetry-preserving pertur-bations, cannot split the band crossing point of H0 atk = 0. That is, the gapless mode at k = 0 described bythe low-energy effective Hamiltonian Heff(k) = λkσx isstable against symmetry-preserving perturbations.

The fact that the Hamiltonian given by Eq.(8) exhibitsa band crossing at k = 0 can directly be seen by comput-ing the eigenstate of G(k) and H(k). Because G(k) andH(k) commute [see Eq.(2)], they can be simultaneouslydiagonalized by the same set of eigenstates

H(k)|±, k〉 = E±(k)|±, k〉, G(k)|±, k〉 = g±(k)|±, k〉,(10)

where the eigenfunctions |±, k〉 are given by

|+, k〉 =1√2

(1

eik2

), |−, k〉 =

1√2

(1

−ei k2

)(11)

3

FIG. 2: (a) Energy spectrum E±(k) of Hamiltonian (9). Blue and orange correspond to the eigenstates E+ and E−, respectively.The two eigenstates are connected smoothly at the boundary of the BZ and cross each other at the center of the BZ. (b)E±(k) as a function of the phases φ of the nonsymmorphic symmetry eigenvalue g±(k). In the space of the eigenvalues g±(k)of the nonsymmorphic symmetry, the two bands are smoothly connected with each other, without any crossing point. (c) kas a function of the phase φ of the eigenvalues g±(k). The two eigenvalue branches are connected at φ = π

2and φ = 3π

2(=

−π2

), leading to a winding number 2. (d) Trajectory of the two bands in the (k, φ, E) space. As a problem of essential three

parameters, the two bands are connected as a circle in the (k, φ, E) space, corresponding to (2, 1) ∈ H1(S1 ×S1 ×R) ∼= Z⊕Z.

and the eigenvalues are

E±(k) = ±2

∞∑n=0

λn sin

(nk +

k

2

), g±(k) = ±e−i k2 .

(12)We find that the energy and nonsymmorphic symme-try eigenvalues of |+, k〉 at k = ±π are continuouslyconnected to the corresponding eigenvalues of |−, k〉 atk = ∓π, see Fig.2. That is, we have{

E+(−π) = E−(π)

E−(−π) = E+(π),

{g+(−π) = g−(π)

g−(−π) = g+(π). (13)

We note that the eigenfunctions |±, k〉 become degen-erate in energy at k = 0 [i.e., E+(0) = E−(0)], whiletheir nonsymmorpic symmetry eigenvalue remains non-degenerate at k = 0 [i.e., g+(0) 6= g−(0)]. Therefore thetwo bands |±, k〉 must cross each other.

To see the topological features of the band structure,we first note that the eigenvalues g±(k) of the nonsym-morphic symmetries G form a manifold as a function ofmomentum k. That is, the eigenvalues g±(k) are mul-tivalued functions of k, with different branches beingsmoothly connected. Inversely, k is a single-valued con-tinuous function of the eigenvalues of the symmetry G.For the two-fold nonsymmorphic symmetry (1), the mo-mentum k ∈ S1 has winding number 2 as a functionof the eigenvalue g±(k) ∈ U(1), which indicates a non-trivial topology [see Fig.2(c)]. To better understand thisnontrivial topology, it is instructive to draw the mutualdependence of the energy eigenvalues E±, the nonsym-morphic eigenvalues g±, and the momentum k in terms ofa trajectory in the three-dimensional space (k, φ, E). Forthe two-band model (9) this is shown in Fig. 2(d). Theprojections of this trajectory onto the three orthogonalplanes (E , k), (E , φ), and (k, φ) are shown in Figs.2(a),2(b) and 2(c), respectively. We can see that the twobands E± are connected as a circle in (k, φ, E) space, cor-responding to the element (2, 1) in the homology groupH1(S1 × S1 × R,Z) ∼= Z⊕ Z.

Instead of P = σ2ı, another possible choice for P isP = σ1ı with the symmetry relations

[H, P ] = 0, PG(k)P−1 = GT (−k), P 2 = 1. (14)

With this choice, we find the following relations for q(k)and qn,

q(k) = q∗(−k), qn = q∗n. (15)

Using Eq. (7), it follows that

q(k) =

∞∑n=0

λn(eink + e−i(n+1)k). (16)

Hence, there always exists a band crossing point at k = π.Let us now show that the algebra obeyed by the sym-

metry operators determines whether the band crossingpoint is at k = 0 or k = π. To that end, we recall thatfor the choice P = σ2ı the operators at the inversion in-variant point k = π, P = σ2ı, G(π) = −iσ2, andH(π) aremutually commuting, see Eq. (6). At the other inversion

invariant point k = 0, however, P = σ2ı and G(0) = σ1

are anti-commuting, while H(0) commutes with P and

G(0), i.e., [H(0), P ] = 0 and [H(0), G(0)] = 0. It followsthat the two degenerate eigenstates of H at k = 0 canbe written as eigenstates of P with different eigenvalues.Explicitly, we find that 1+i

2 |+, 0〉+1−i

2 |−, 0〉 is an eigen-

state of P with eigenvalue +1, while 1−i2 |+, 0〉+

1+i2 |−, 0〉

is an eigenstate of P with eigenvalue −1. Therefore, theband crossing, which is protected by P , occurs at k = 0.

A similar analysis can be performed for the choiceP = σ1ı, i.e., the Hamiltonian given by Eq. (16). Inthat case, we find that at k = 0 the operators H(0),

G(0), and P are mutually commuting, while P and G(k)anti-commute at k = π, where the band degeneracy islocated. We conclude that the algebraic relations obeyedby the symmetry operators determine the location of thesymmetry-enforced band crossing, see Table I.

4

IV. NONSYMMORPHIC MAGNETICSYMMETRY

From the discussion in the previous section it followsthat not all the symmetry constraints are necessary to en-force the existence of the band crossing. As we shall see,a single nonsymmorphic antiunitary symmetry, namely amagnetic nonsymmorphic symmetry, is sufficient to en-sure the existence of a band crossing at k = 0 or k = π.As illustrated in Fig. 1(b), a magnetic nonsymmorphicsymmetry can be viewed as the combination of a nonsym-morphic symmetry G(k) with a time-reversal symmetry

T . We only require that the combined symmetry GT issatisfied. That is, both G and T may be broken individ-ually, but the combination must be preserved. In whatfollows we assume that T 2 = +1 and consider two possi-ble choices for T , namely, (i) T = Kı and (ii) T = σ3Kı,

where K denotes the complex conjugation operator. Byuse of Eq. (1), we find that in case (i) [T , G(k)] = 0, while

in case (ii) {T , G(k)} = 0.Let us start with the discussion of case (i), where the

combined symmetry G = GT = G(k)Kı acts on H(k) as

[G(k),H(k)] = 0. (17)

Inserting Eq.(3) into Eq. (17), one obtains eikq(k) =q(−k), and qn = q−(n+1) by use of q(k) =

∑n qne

ink,which implies

q(k) =

∞∑n=0

qn(eink + e−i(n+1)k). (18)

We observe that due to the symmetry constraint Eq. (17),q(π) = 0, i.e., there is a band crossing point at k =π. Note that the term fz(k)σ3 is not forbidden by the

combined symmetry GT . However, Eq. (17) requires thatfz(k) is an odd function of k. Hence fz(k)σ3 must vanishat the high-symmetry points k = 0 and k = π, due tothe periodicity of the BZ, i.e., fz(k) = fz(k + 2π). To

summarize, the magnetic nonsymmorphic symmetry GTis sufficient to enforce the existence of a band crossing atk = π.

Next, we discuss choice (ii) for T , in which case the

combined symmetry G′ ≡ GT = Gσ3Kı acts on H(k) as

[G′(k),H(k)] = 0. (19)

Combining Eq. (3) with Eq. (19) we find that eikq(k) =−q(−k) . Hence, the Fourier components qn must satisfyqn = −q−(n+1), which yields

q(k) =

∞∑n=0

qn(eink − e−i(n+1)k). (20)

Since q(0) = 0 in Eq. (20), there is an unavoidable bandcrossing at k = 0. As before, the term fz(k)σ3 is symme-try allowed, with fz(k) an odd function. Hence, fz(k)σ3

Position G,P GT

k = 0 PG = −G†P {G, T} = 0

k = π PG = G†P [G, T ] = 0

TABLE I: The positions of the band crossings in the BZ aredetermined by the algebra of the symmetry operators.

vanishes at the high-symmetry points k = 0 and k = π,and therefore cannot gap out the band crossing point.

From the above discussion, one infers that the com-mutation relation between T and G(k) determines theposition of the band-degeneracy points. Namely, for[G, T ] = 0 [case (i)] and {G, T} = 0 [case (ii)], we have

(GT )2 = +e−ikσ0 and (GT )2 = −e−ikσ0, respectively.

Hence, we find that for case (i) (GT )2 = −1 at k = π,

while for case (ii) (GT )2 = −1 at k = 0. Since GT is

an anti-unitary operator, (GT )2 = −1 leads to a banddegeneracy, in analogy to Kramers theorem. Thus, for[G, T ] = 0 the band-crossing point is at k = π, while for

{G, T} = 0 it is at k = 0, see Table I.We note that a number of recent works16,17,21,29 have

discussed edge band structures of two-dimensional (2D)nonsymmorphic insulators that are similar to the 1D bulkband structures studied here. In contrast to conven-tional topological insulators, the edge bands of these 2Dnonsymmorphic insulators do not connect valence andconduction bands. Hence, our results for the bulk bandstructures of 1D systems, can be applied directly to theedge spectrum of these 2D nonsymmorphic insulators.This suggests, in particular, that the crossing of the edgebands of these 2D systems is, at least in some cases, en-forced by the nonsymmorphic symmetry of the edge the-ory.

In closing this section, we note that the existence of aband crossing cannot be enforced by a nonsymmorphicparticle-hole symmetry, which is discussed in detail inAppendix A.

V. TOPOLOGICAL CLASSIFICATION OFBAND-CROSSING POINTS

Let us now derive the classification of the global topo-logical properties of the considered Hamiltonians. Byglobal topology, we mean a band structure is allowedto be deformed smoothly in the whole momentum spacewith the symmetries being preserved, in contrast to theordinary local one, where deformations are restrictedonly in a open neighborhood around the band-crossingpoint. The group structure is given by the direct sumof the Hamiltonians. To that end, we study whether theband crossings of the doubled Hamiltonians H ⊗ τ0 andH⊗τ3 can be gapped out by symmetry-preserving terms.Here, τµ’s represent an additional set of Pauli matricesand τ0 is the 2× 2 identity matrix. The symmetry oper-ators for the doubled Hamiltonians are G(k)⊗τ0, P ⊗τ0,

and T ⊗ τ0. We observe that diag(H, λH) can be contin-

5

uously deformed to diag(H,−λH) without breaking thesymmetries and without opening a gap at k = 0 or π. Inaddition, we find that mσ0⊗τ1 is a symmetry preservingmass term that gaps out the spectrum of H ⊗ τ3 in theentire BZ. It follows that the global topological featuresof H possess a Z2 classification, namely even number ofcopies can be gapped with symmetries being preserved,while odd number cannot. It is noted that although inthe above simple deformation the gap is fully closed atλ = 0, a more carefully chosen deformation can be madesuch that at any intermediate stage band crossing hap-pens only at finite number of momenta.

To infer the classification of the local topological prop-erties of the band crossing points, we enclose the degen-eracy point by an S0 sphere (consisting of two pointson the left and right of the degeneracy point) and con-sider adiabatic deformations that do not close the gapon the chosen S0. The only possible gap opening term isfz(k)σ3, which however vanishes at the high-symmetrypoints k = 0, π due to the nonsymmorphic symmetry.This also holds for multiple copies of H. From this weconclude that the local topological features of the bandcrossing points exhibit a Z classification.

VI. HIGHER-DIMENSIONALGENERALIZATIONS

Our results for symmetry-enforced band crossings in1D can be readily generalized to higher dimensions. Weassume that the fractional translation is along the kxdirection for d-dimensional systems. The d-dimensionalHamiltonian H(k) can then be decomposed into a fam-ily of 1D Hamiltonians hk⊥(kx) = H(kx, k⊥), which areparametrized by the (d − 1) momenta k⊥ perpendicu-lar to kx. Let us briefly discuss how the three differenttypes of nonsymmorphic symmetries that we consideredabove constrain this d-dimensional Hamiltonian. (i) IfH(k) is invariant under a unitary nonsymmorphic sym-metry G(kx), then there are in general several branchesof Fermi surfaces (possibly of dimension d > 0) that areparametrized by k⊥. (ii) If there exists in addition areflection symmetry reversing kx, then the Fermi sur-faces are pinned at kx = 0 or kx = π, depending onthe algebraic relations obeyed by the symmetry opera-tors, as specified in Table I. (iii) If we consider symme-tries that relate k to −k, such as a nonsymmorphic mag-netic symmetry (or an additional inversion), then thereexist 2d−1 1D inversion invariant subsystems hka⊥(kx)

(a = 1, · · · , 2d−1) of hk⊥(kx) , which are labeled bythe perpendicular momenta ka⊥ that are invariant underk⊥ → −k⊥. These subsystems have band degeneraciesat kx = 0 or kx = π, as determined by the algebraic rela-tions in Table I. The other 1D subsystems hk⊥ , where k⊥is not invariant under k⊥ → −k⊥, are generally gapped.

(I)

(II)

FIG. 3: Illustration of the SSH model. (a) For α 6= 0 the SSHmodel breaks the nonsymmorphic symmetry G(k), Eq. (1).(b) At the critical point α = 0 the translation symmetry ispromoted from 2Z to Z, such that the nonsymmorphic sym-metry G(k) is satisfied. (c) Energy spectrum at α = 0 in (I)the original BZ and (II) the unfolded BZ.

A. Examples

We illustrate our theoretical results by two examples:(i) The Su-Schrieffer-Heeger (SSH) model30 and (ii) theπ-flux square lattice model. The Hamiltonian of the SSHmodel is given by

HSSH(k) =

(0 ∆(k)

Ơ(k) 0

), (21)

where ∆(k) = (t + α) + (t − α)e−ik, see Fig. 3(a). Atthe critical point α = 0 the system is invariant under ahalf translation followed by an exchange of A and B sub-lattices, which corresponds to the nonsymmorphic sym-metry G(k), Eq. (1), see Fig. 3. Since the SSH model

also has the inversion symmetry P = σ1ı, we find in ac-cordance with Table I that there is a gapless point atk = π. Observe that at α = 0 the translation sym-metry is promoted from 2Z (with translator 2a) to Z(with the translator a) and the SSH model becomes a 1Dtight-binding model of free fermions with the dispersionE(k) = 2t cos(k) [see Fig. 3(c)]. It is noted that in orderto create a bandcrossing point protected by nonsymmor-phic symmetry in a tight-binding model through dimer-ization, one must ensure that the original non-dimerizedmodel has two chiral gapless modes. A nonvanishingα, on the other hand, reduces the translation symmetryfrom Z to 2Z and breaks the nonsymmorphic symmetryG(k), leading to a topological (α < 0) or trivial (α > 0)insulator, depending on the sign of α.

The Hamiltonian of the π-flux square lattice modelreads in momentum space

H(k) =

(2t sin kx t+ te−iky

t+ teiky −2t sin kx

), (22)

where t denotes the nearest neighbor hopping amplitude,see Fig. 4(a). The model is invariant under the non-

symmorphic magnetic symmetry G(ky)Ki, which corre-

sponds to a time-reversal symmetry T = Kı followed by

6

FIG. 4: (a) Illustration of the π-flux square lattice model.This model is invariant under the nonsymmorphic magneticsymmetry G(ky)Ki, namely a time-reversal symmetry fol-lowed by the nonsymmorphic symmetry G(ky). (b) Energyspectrum of the π-flux square lattice model.

a half translation along y and an exchange of A and Bsublattices. The high-symmetry 1D subsystems kx = 0and kx = π have gapless Dirac points at ky = π, asshown in Fig. 4(b). This is in agreement with Table I,

since [G(ky), Ki] = 0. Similar to the SSH model, theπ-flux state is driven into a topological or trivial insu-lating phase by a dimerization α along y, that breaksthe nonsymmorphic symmetry G(ky). In closing, we ob-serve that the π-flux model can be viewed as a higher-dimensional generalization of the SSH model.

VII. DISCUSSIONS ABOUT MULTI-BANDTHEORIES

It is noted that the topological arguments in this workare not limited to two-band models, which is exemplifiedby two cases in follows. First, if a multi-band theory hasa chiral symmetry, then the Hamiltonian can be anti-diagonalized with the upper-right entry being a matrix∆(k), and equation (4) still holds for q(k) = Det(∆(k)).The topological argument around Eq.(5) implies thatDet(∆(k)) has to vanish somewhere in momentum space,namely there exists band-crossing points enforced by thechiral and nonsymmorphic symmetry.

Secondly, let us extend the spinless time-reversal sym-metry discussed in Sec.IV to the spinful one T = −iσ2Kı,which acts in spinful four-band theories. If both T andG are preserved (G acts on the space of τ), equations(4) and (18) hold for q(k) = Det(∆(k)), which impliesDet(∆(k)) vanishes at k = π. But diagonal terms haveto vanish at k = 0 and π as required by the symme-tries, except the chemical potential term. Thus bandsare enforced to cross at k = π. Note that the vanishingof Det(∆(k)) at k = π needs only the combined symme-

try GT , but both T and G are required for that of thediagonal terms.

Acknowledgments

The authors thank C.-K. Chiu for useful discussions.

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Appendix A: Nonsymmorphic Particle-holesymmetry

In this section, we discuss whether the existence of aband crossing can also be enforced by a nonsymmorphicparticle-hole symmetry. As it turns out this is not possi-ble. To see this, we consider for instance an anti-unitarynon-symmorphic particle-hole symmetry G′′ = G(k)K,which acts on H as

G(k)H(k)G−1(k) = −H∗(−k). (A1)

Hence, the off-diagonal component of H(k) must satisfyeikq(k) = −q(−k), and its Fourier components obey qn =−q−(n+1). Therefore, q(k) can be written as

q(k) =

∞∑n=0

qn(eink − e−i(n+1)k), (A2)

which vanishes at k = 0. However, there does not exist asymmetry protected band crossing at k = 0, since the gapopening term z(k)σ3, with z(k) = z(−k) an even func-tion, preserves the non-symmorphic particle-hole symme-try G′′. To protect the band crossing point at k = 0, anadditional symmetry is needed which forbids the z(k)σ3

term. For example, the chiral symmetry {H, S} = 0,

with S = σ3 and {S, G′′} = 0, prevents the z(k)σ3 term.